Editing Gram matrix (section)

==Properties==
===Positive-semidefiniteness===
The Gram matrix is [[symmetric matrix|symmetric]] in the case the inner product is real-valued; it is [[Hermitian matrix|Hermitian]] in the general, complex case by definition of an [[inner product]].

The Gram matrix is [[Positive-semidefinite matrix|positive semidefinite]], and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can be seen from the following simple derivation:
: <math>
  x^\dagger \mathbf{G} x =
  \sum_{i,j}x_i^* x_j\left\langle v_i, v_j \right\rangle =
  \sum_{i,j}\left\langle x_i v_i, x_j v_j \right\rangle =
  \biggl\langle \sum_i x_i v_i, \sum_j x_j v_j \biggr\rangle =
  \biggl\| \sum_i x_i v_i \biggr\|^2 \geq 0 .
</math>

The first equality follows from the definition of matrix multiplication, the second and third from the bi-linearity of the [[inner-product]], and the last from the positive definiteness of the inner product.
Note that this also shows that the Gramian matrix is positive definite if and only if the vectors <math> v_i </math> are linearly independent (that is, <math display="inline">\sum_i x_i v_i \neq 0</math> for all <math>x</math>).<ref name="HJ-7.2.10"/>

===Finding a vector realization===
{{See also|Positive definite matrix#Decomposition}}
Given any positive semidefinite matrix <math>M</math>, one can decompose it as:
: <math>M = B^\dagger B</math>,

where <math>B^\dagger</math> is the [[conjugate transpose]] of <math>B</math> (or <math>M = B^\textsf{T} B</math> in the real case).

Here <math>B</math> is a <math>k \times n</math> matrix, where <math>k</math> is the [[matrix rank|rank]] of <math>M</math>. Various ways to obtain such a decomposition include computing the [[Cholesky decomposition]] or taking the [[square root of a matrix|non-negative square root]] of <math>M</math>.

The columns <math>b^{(1)}, \dots, b^{(n)}</math> of <math>B</math> can be seen as ''n'' vectors in <math>\mathbb{C}^k</math> (or ''k''-dimensional Euclidean space <math>\mathbb{R}^k</math>, in the real case). Then
: <math>M_{ij} = b^{(i)} \cdot b^{(j)}</math>

where the [[dot product]] <math display="inline">a \cdot b = \sum_{\ell=1}^k a_\ell^* b_\ell</math> is the usual inner product on <math>\mathbb{C}^k</math>.

Thus a [[Hermitian matrix]] <math>M</math> is positive semidefinite if and only if it is the Gram matrix of some vectors <math>b^{(1)}, \dots, b^{(n)}</math>. Such vectors are called a '''vector realization''' of {{nowrap|<math>M</math>.}} The infinite-dimensional analog of this statement is [[Mercer's theorem]].

===Uniqueness of vector realizations===
If <math>M</math> is the Gram matrix of vectors <math>v_1,\dots,v_n</math> in <math>\mathbb{R}^k</math> then applying any rotation or reflection of <math>\mathbb{R}^k</math> (any [[orthogonal transformation]], that is, any [[Euclidean isometry]] preserving 0) to the sequence of vectors results in the same Gram matrix. That is, for any <math>k \times k</math> [[orthogonal matrix]] <math>Q</math>, the Gram matrix of <math>Q v_1,\dots, Q v_n</math> is also {{nowrap|<math>M</math>.}}

This is the only way in which two real vector realizations of <math>M</math> can differ: the vectors <math>v_1,\dots,v_n</math> are unique up to [[orthogonal transformation]]s. In other words, the dot products <math>v_i \cdot v_j</math> and <math>w_i \cdot w_j</math> are equal if and only if some rigid transformation of <math>\mathbb{R}^k</math> transforms the vectors <math>v_1,\dots,v_n</math> to <math>w_1, \dots, w_n</math> and 0 to 0.

The same holds in the complex case, with [[unitary transformation]]s in place of orthogonal ones.
That is, if the Gram matrix of vectors <math>v_1, \dots, v_n</math> is equal to the Gram matrix of vectors <math>w_1, \dots, w_n</math> in <math>\mathbb{C}^k</math> then there is a [[unitary matrix|unitary]] <math>k \times k</math> matrix <math>U</math> (meaning <math>U^\dagger U = I</math>) such that <math>v_i = U w_i</math> for <math>i = 1, \dots, n</math>.<ref>{{harvtxt|Horn|Johnson|2013}}, p. 452, Theorem 7.3.11</ref>

===Other properties===
* Because <math>G = G^\dagger</math>, it is necessarily the case that <math>G</math> and <math>G^\dagger</math> commute.  That is, a real or complex Gram matrix <math>G</math> is also a [[normal matrix]].
* The Gram matrix of any [[orthonormal basis]] is the identity matrix.  Equivalently, the Gram matrix of the rows or the columns of a real [[rotation matrix]] is the identity matrix.  Likewise, the Gram matrix of the rows or columns of a [[unitary matrix]] is the identity matrix.
* The rank of the Gram matrix of vectors in <math>\mathbb{R}^k</math> or <math>\mathbb{C}^k</math> equals the dimension of the space [[Linear span|spanned]] by these vectors.<ref name="HJ-7.2.10"/>